Testing for Weak Instruments
To test for weak instruments, regress each independent variable suspected of being contemporaneously correlated with the error (xk) onto all of the instruments (internal and external). Suppose xK is the endogenous regressor. The first stage regression is:
xk = Yi + Y2X2 +—– + Y к-1XK-1 + вігі +——- + Olzl + vk (10.5)
In this notation, the z1, …, zl are the external instruments. The others, x2, …, zk-1 are exogenous and are used as instruments for themselves (i. e., internal to the model). If the F – statistic associated with the hypothesis that the coefficients on the external instruments, в1, …, eL are jointly zero is less than 10, then you conclude that the instruments are weak. If it is greater than 10, you conclude that the instruments are strong enough. The following script uses least squares to perform three such tests. The first regression assumes there is only one instrument, z1; the second that the single instrument is z2; the third assumes both are instruments.
open "@gretldirdatapoemroz. gdt smpl wage>0 —restrict
list x = const educ exper sq_exper
list z2 = const exper sq_exper mothereduc fathereduc ols educ z2
omit mothereduc fathereduc
When omit follows an OLS regression, gretl estimates a restricted model where the variables listed after it are omitted from the model above. It then performs a joint hypothesis test that the coefficients of the omitted variables are zero against the alternative that one or more are not zero. The —quiet option reduces the amount of output you have to wade through by suppressing the regressions; only the test results are printed. The output from gretl appears in Figure 10.3 below: Since the F value = 55.4, which is well beyond 10. We reject the hypothesis that the (external) instruments mothereduc and fathereduc are weak in favor of the alternative that they are strong.